We establish the existence of multiple positive solutions of nonlinear equations of the form \begin{equation*} -u''(t)=g(t)f(t,u(t)),\ t \in (0,1), \end{equation*} where $g, f$ are non-negative functions, subject to various nonlocal boundary conditions. The common feature is that each can be written as an integral equation, in the space $C[0,1]$, of the form $$ u(t)=\gamma(t){\alpha}[u]+\int_{0}^{1}k(t,s)g(s)f(s,u(s))\,ds $$ where $\alpha[u]$ is a linear functional given by a Stieltjes integral but is \emph{not} assumed to be positive for all positive $u$. Our new results cover many nonlocal boundary conditions previously studied on a case by case basis for particular positive functionals only, for example, many $m$-point BVPs are special cases. Even for positive functionals our methods give improvements on previous work. Also we allow weaker assumptions on the nonlinear term than were previously imposed.

Positive solutions of nonlocal boundary value problems involving integral conditions

INFANTE, GENNARO
2008-01-01

Abstract

We establish the existence of multiple positive solutions of nonlinear equations of the form \begin{equation*} -u''(t)=g(t)f(t,u(t)),\ t \in (0,1), \end{equation*} where $g, f$ are non-negative functions, subject to various nonlocal boundary conditions. The common feature is that each can be written as an integral equation, in the space $C[0,1]$, of the form $$ u(t)=\gamma(t){\alpha}[u]+\int_{0}^{1}k(t,s)g(s)f(s,u(s))\,ds $$ where $\alpha[u]$ is a linear functional given by a Stieltjes integral but is \emph{not} assumed to be positive for all positive $u$. Our new results cover many nonlocal boundary conditions previously studied on a case by case basis for particular positive functionals only, for example, many $m$-point BVPs are special cases. Even for positive functionals our methods give improvements on previous work. Also we allow weaker assumptions on the nonlinear term than were previously imposed.
2008
nonlocal boundary conditions; fixed point index; positive solution
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11770/128560
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